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Properties

Label 2-201-67.29-c3-0-23
Degree $2$
Conductor $201$
Sign $0.995 + 0.0912i$
Analytic cond. $11.8593$
Root an. cond. $3.44374$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.32 + 4.02i)2-s + 3·3-s + (−6.80 − 11.7i)4-s + 1.94·5-s + (−6.97 + 12.0i)6-s + (−1.41 − 2.45i)7-s + 26.0·8-s + 9·9-s + (−4.50 + 7.81i)10-s + (−29.5 − 51.1i)11-s + (−20.4 − 35.3i)12-s + (−26.7 + 46.4i)13-s + 13.1·14-s + 5.82·15-s + (−6.12 + 10.6i)16-s + (48.4 − 83.8i)17-s + ⋯
L(s)  = 1  + (−0.821 + 1.42i)2-s + 0.577·3-s + (−0.850 − 1.47i)4-s + 0.173·5-s + (−0.474 + 0.821i)6-s + (−0.0766 − 0.132i)7-s + 1.15·8-s + 0.333·9-s + (−0.142 + 0.247i)10-s + (−0.808 − 1.40i)11-s + (−0.490 − 0.850i)12-s + (−0.571 + 0.990i)13-s + 0.251·14-s + 0.100·15-s + (−0.0956 + 0.165i)16-s + (0.691 − 1.19i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 + 0.0912i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.995 + 0.0912i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(201\)    =    \(3 \cdot 67\)
Sign: $0.995 + 0.0912i$
Analytic conductor: \(11.8593\)
Root analytic conductor: \(3.44374\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{201} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 201,\ (\ :3/2),\ 0.995 + 0.0912i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.945696 - 0.0432149i\)
\(L(\frac12)\) \(\approx\) \(0.945696 - 0.0432149i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 3T \)
67 \( 1 + (471. - 279. i)T \)
good2 \( 1 + (2.32 - 4.02i)T + (-4 - 6.92i)T^{2} \)
5 \( 1 - 1.94T + 125T^{2} \)
7 \( 1 + (1.41 + 2.45i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (29.5 + 51.1i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (26.7 - 46.4i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + (-48.4 + 83.8i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-27.3 + 47.3i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-87.8 + 152. i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (77.2 + 133. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (-96.1 - 166. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (-47.4 + 82.1i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (47.3 + 81.9i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 - 29.3T + 7.95e4T^{2} \)
47 \( 1 + (-255. - 442. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + 271.T + 1.48e5T^{2} \)
59 \( 1 - 115.T + 2.05e5T^{2} \)
61 \( 1 + (136. - 236. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
71 \( 1 + (370. + 641. i)T + (-1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 + (-503. + 872. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-55.8 - 96.7i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (-697. + 1.20e3i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + 537.T + 7.04e5T^{2} \)
97 \( 1 + (-376. + 652. i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.95175508618059325876713160489, −10.60909531632803735864915953935, −9.507625669406649505770673236033, −8.876623443103124188714646296681, −7.87427276571907831021455727032, −7.10469157246353134255375482093, −5.99426200474061189625320731823, −4.83817300052630167147267223947, −2.85271489320924308337051889658, −0.51323939950625920850364769541, 1.52173307371852343711817545218, 2.64013982389583318899253487243, 3.79390997988469794230871853093, 5.45984429402594974590380924093, 7.52607169632082126341064100554, 8.143940988889327497833618980487, 9.530926657029251702475519511387, 9.933382955420917552400970158134, 10.75628551457485797592219343443, 12.06042774786482105869064319748

Graph of the $Z$-function along the critical line